Place the kiwi in a large bowl with sugar and lemon juice. Stir thoroughly and slightly mash the kiwi with the sugar.
Cover and place in the refrigerator to macerate (soften and form syrup) for anywhere between 2 and 48 hours (I typically do this step overnight). Place several small plates in the freezer to test the set later.
Place the fruit mixture into a strainer over your cooking pot and allow the syrup to drip for at least 20 minutes, stirring occasionally to release additional syrup. Place the strainer of fruit over a bowl to catch any additional syrup. This is a good time to start sterilizing your jars and lids in a water bath if you haven't yet done so.
Turn the heat of your cooking pot on high, clip on a candy thermometer, and bring the syrup to a steady boil. The mixture will foam quite heavily so stir well and often, monitoring to prevent boil over. When the foam clears at 220℉ (approximately 8-10 minutes) add the fruit from the colander and any additional syrup from the catch bowl and bring to a foamy boil (10-20 minutes depending on quantity of fruit). Continue stirring frequently to prevent burning.
As you boil, watch for the preserves to begin to set. There will be a hardy boil that will not stir down (will continue to boil vigorously as your stir), you may hear popping along the bottom of the pot, the bubbles will be more gel-like and syrupy, and any fruit chunks that were previously floating should now sink in the syrup mixture. When the foam has almost completely cleared remove the pot from heat and let sit for a couple minutes before spooning some onto a frozen plate to test the set (see notes). If needed, return to heat and continue cooking on high in 2-3 minute increments.
Stir well and ladle preserves into the jars leaving a ½ inch headspace. Clean the rims of the jars well and cover, finger-tightening the rings.
Lover the jars upright into the boiling water of the water-bath canner, having the jars covered by at least 1 inch of water. Process for 10 minutes starting from when the water comes back to a vigorous boil.
Remove jars from the canner and allow to cool, listening for pings to indicate the jars have sealed. Allow to cool for 12 hours before testing seals, washing jars, labeling, and storing.